368 research outputs found
Self-dual cyclic codes over finite chain rings
Let be a finite commutative chain ring with unique maximal ideal , and let be a positive integer coprime with the
characteristic of . In this paper, the algebraic
structure of cyclic codes of length over is investigated. Some new
necessary and sufficient conditions for the existence of nontrivial self-dual
cyclic codes are provided. An enumeration formula for the self-dual cyclic
codes is also studied.Comment: 15 page
Computing sparse multiples of polynomials
We consider the problem of finding a sparse multiple of a polynomial. Given f
in F[x] of degree d over a field F, and a desired sparsity t, our goal is to
determine if there exists a multiple h in F[x] of f such that h has at most t
non-zero terms, and if so, to find such an h. When F=Q and t is constant, we
give a polynomial-time algorithm in d and the size of coefficients in h. When F
is a finite field, we show that the problem is at least as hard as determining
the multiplicative order of elements in an extension field of F (a problem
thought to have complexity similar to that of factoring integers), and this
lower bound is tight when t=2.Comment: Extended abstract appears in Proc. ISAAC 2010, pp. 266-278, LNCS 650
Cyclotomic numerical semigroups
Given a numerical semigroup , we let be its semigroup polynomial. We study cyclotomic numerical semigroups;
these are numerical semigroups such that has all its roots
in the unit disc. We conjecture that is a cyclotomic numerical semigroup if
and only if is a complete intersection numerical semigroup and present some
evidence for it. Aside from the notion of cyclotomic numerical semigroup we
introduce the notion of cyclotomic exponents and polynomially related numerical
semigroups. We derive some properties and give some applications of these new
concepts.Comment: 17 pages, accepted for publication in SIAM J. Discrete Mat
Idempotents in Cyclic Codes
Cyclic codes give us the most probable method by which we may detect and correct data transmission errors. These codes depend on the development of advanced mathematical concepts. It is shown that cyclic codes, when viewed as vector subspaces of a vector space of some dimension n over some finite field F, can be approached as polynomials in a ring. This approach is made possible by the assumption that the set of codewords is invariant under cyclic shifts, which are linear transformations. Developing these codes seems to be equivalent to factoring the polynomial x[superscript]n-x over F. Each factor then gives us a cyclic code of some dimension k over F. Constructing factorizations of x[superscript]n-x is accomplished by using cyclotomic polynomials and idempotents of the code algebra. The use of these two concepts together allows us to find cyclic codes in F[superscript]n. Hence, the development of cyclic codes is a journey from codewords and codes to fields and rings and back to codes and codewords.M.A
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